Nonlinear hyperbolic smoothing at a focal point (Q5954530)

The authors construct real valued finite energy solutions of the dissipative nonlinear wave equation NEWLINE\[NEWLINE(\partial^2_t- \Delta)u+|u |^{h-1} u=0,\;1<h\in \mathbb{R}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0,x)= f(x),\;u_t(0,x)=g(x),NEWLINE\]NEWLINE where it is assumed that the initial data \((f,g)\) are piecewise \(C^2\), radial, compactly supported, vanish for \(|x|\leq 1\), and have singularity only on \(|x|=1\). In addition, \(f\) is assumed to be continuous, \(g+ \partial_{ |x|}f\) is not continuous but \(g-\partial_{|x|}f\) is continuous at \(|x|=1\). Then it can be proved that \((u,u_t)\in L^\infty( [0,1): H^\sigma (\mathbb{R}^d)\times H^{\sigma-1} (\mathbb{R}^d))\) if and only if \(\sigma <3/2\), that if \(d> 2h/(h-1)\), then NEWLINE\[NEWLINEu\in L^\infty \bigl([1,\infty) \bigr): H^2(\mathbb{R}^d),\;u_t\in L^\infty\bigl( [1,\infty) \bigr): H^1(\mathbb{R}^d)NEWLINE\]NEWLINE and that, if \(2h/(h-1)- 1<d\leq 2h/ (h-1)\), then NEWLINE\[NEWLINEu\in C\bigl( [1,\infty): H^{2-\sigma- \varepsilon}(\mathbb{R}^d) \bigr),\;u_t\in C\bigl([1, \infty):H^{1-\sigma- \varepsilon}(\mathbb{R}^d) \bigr).NEWLINE\]